= Solution
When the imposed <magnetic field> vanishes, the dynamical <dispersion relation> becomes
$$
\sigma^2\left[\sigma^2+(N^2+\Omega^2)\right]=0.
$$
The nonstationary hydrodynamic branch therefore has $\sigma^2=-(N^2+\Omega^2)$. \b[The <radial Solberg–Høiland instability criterion>] in this local model is
$$
\boxed{N^2+\Omega^2<0.}
$$
The <radial buoyancy frequency> supplies either restoration or a driving force, while the <radial epicyclic frequency> $\kappa=\Omega$ supplies rotational restoration. Zero sum is marginal, and positive sum gives stable oscillations. The neutral roots do not themselves signify exponential growth.
Because both equilibrium <pressure> and dimensionless <specific entropy> decrease outward, their radial gradients have the same sign. The leading minus sign in the expression for $N^2$ therefore makes $N^2<0$: the radial stratification is adverse. For smooth profiles varying over <radius> $r$, $|\partial_rP|\sim P/r$ and $|\partial_rS|\sim1/r$, so
$$
N^2\sim-\frac{P}{\rho r^2}\sim-\frac{c_s^2}{r^2}\sim-\left(\frac Hr\right)^2\Omega^2.
$$
For a <thin disk>, the magnitude of this negative <radial buoyancy frequency> squared is much less than $\Omega^2$. \b[Rotation stabilizes the hydrodynamic mode despite the adverse entropy gradient.] This estimate assumes gradients on the global radial scale; a sharp thermal feature may be different. It addresses the ideal nondiffusive equations here. Processes such as <convective overstability> require additional thermal relaxation and are not ruled out by this particular criterion.
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